Melt butter in a medium-size heavy-bottomed skillet on medium-high heat.

When butter just begins to brown, add onions, salt, and pepper. Stir periodically until onions soften, about ten minutes.

Add balsamic vinegar and stir rapidly to coat the onions. Continue to stir periodically until onions are caramelized, about five minutes. Remove onions from the skillet and set aside. Return cleaned skillet (or use a second heavy-bottomed skillet) to the stovetop set to medium heat.

To assemble quesadillas, place about two tablespoons of cheese on half of each tortilla. Top cheese with two tablespoons of the cooked onions. In a single layer over the onions, alternately arrange two apple slices and two avocado slices. Top with another two tablespoons of cheese. Fold each tortilla in half.

Cook as many quesadillas as will fit in a single layer in the skillet for about 60 seconds (or until the bottoms are just brown). Then, flip the quesadillas and cook the other sides for another 60 seconds. Repeat until all quesadillas are cooked.